Definition of Linear Momentum

V m P

=So what?

– collisions– explosions

Conservation of momentum helps us solve certain types of problems

Newton’s Second Law:New version

“The rate of change of momentum of an object is equal to the net force applied to it”

ΣF = dp/dt If we exert a force on a body, the momentum

of the body changes

Momentum of a closed system

is conserved

∑∑ =i

'i

'i

iii vm vm

BEFORE and AFTER

Momentum is ALWAYS conserved for a SYSTEM, you just have to look

at a big enough system to see it correctly.

CollisionsMomentum before the collision is equal to momentum after the collision.

all vectors!!!

vm vm

vm vm'22

'11

2211

+

=+

Definitions:• Elastic collision = kinetic

energy is conserved• Inelastic collision =

kinetic energy is not conserved.

Momentum conserved? Total Energy conserved?

Example: Colliding TrainsThe train car on the left, mass m1, is moving with speed Vowhen it collides with a stationary car of mass m2. The two stick together.

1. What is their speed after the collision?2. Show that the collision is inelastic.

Elastic collisions

Another example:

A ball of mass m1 collides head on (elastically) with a second ball at rest and rebounds (goes in the opposite direction) with speed equal to ¼ of its original speed. What is the mass of the second ball m2?

Bottom line: You use momentum conservation• When you don’t know the forces

in the system• When you are studying all of the

pieces of the system which are producing forces

Ballistic PendulumA bullet of mass m and velocity Voplows into a block of wood with mass M which is part of a pendulum.

How high does the block of wood go?

Is the collision elastic or inelastic?

SolutionStep 1: The bullet plows into a blockUse momentum conservation:

m vo = (m + M) v ’v ’ = m vo / (m + M)

Step 2: The pendulum swings upUse energy conservation:

(m + M) v ’2 / 2 = (m + M) g h

h = v ’2 / (2 g) = (m vo)2 / [(m + M)2 2 g]

Car collisions in 2 D

Solv

ing

thos

e 3

equa

tions

(from

the

prev

ious

slid

e):

2 D billiards

Center of Mass (CM) motion

What else is CM good for?

• Two ways for solving collision and explosion problems (they are the same laws of physics)– Conservation of Momentum – Watching the Center of Mass

Use whichever is easier

ExampleA two stage rocket is on a trajectory. At the peak it has traveled a distance d and it breaks into two equal mass pieces. Part I falls straight down with no initial velocityWhere does the 2nd half of the rocket end up?

CM by IntegrationA rod of length Land mass M has a uniform density.

Calculate the center of mass.

What if the density varied linearly from 0 to some value at the end?